analytical modeling of characteristic maps of the sr-30

Analytical Modeling of Characteristic Maps of the

SR-30 Turbojet Engine

Joaquín Valencia Bravo*, Frederick Just Agosto,

David Serrano Acevedo, Marco Menegozzo Department of Mechanical Engineering

University of Puerto Rico -Mayagüez

Mayagüez, Puerto Rico

Orlando Ruiz Quinones

Writing Systems Engineer, Hewlett Packard Inc.

Aguadilla, Puerto Rico

Abstract—A crucial step in the mathematical modeling of a

gas turbine engine is the characteristic map development of its

main components, the compressor and the turbine. Maps can

be obtained from manufacturers, although they are generally

not available to the user. Furthermore, the construction of

these maps through experiments is expensive. Analytical

models prove to be economically more advantageous.

Compared to other methods, they require little experimental

data to be validated. These models are based on physical

principles, so the results are more accurate compared to those

obtained through numerical simulations or the use of neural

networks. This approach allows more flexibility to make

changes to the performance characteristics of engine

components. These changes may be achieved by modifying

certain geometric measurements, this quality could be used in

diagnostic and/or engine control systems. In this work the

characteristic map of the SR-30 turbojet engine components is

analytically modeled. Geometric dimensions and properties of

the working fluid were taken as data. This is an one-

dimensional model in which it is analyzed the aerodynamics

and thermodynamics parameters in each station of the

centrifugal compressor and axial turbine. Design point

calculations are used to estimate the values of working fluid

parameters such as velocity, pressure, temperature, enthalpy,

and entropy function. Results of this work are compared with

those obtained by other authors. For greater reliability in

these results, an experimental validation is necessary.

Keywords — Turbojet engine; characteristic map; analytical

model; compressor; turbine

I. INTRODUCTION

Turbomachinery component performance maps constitute

an essential input to the development of engine

performance models. Maps may be obtained from

experimental procedures, but this can result in an expensive

endeavor. Furthermore, experiments might not completely

describe the full range of interest, and the use of

extrapolationis required. Manufacturers usually

developproduct component characteristic maps but this

information is rarely distributed to customers. In addition

to experimental methods, there are other alternatives in

obtaining performance maps. Kurzke [1] used data from

published maps to reproduce component maps through

commercial programs such as Smooth_C [2] and

Smooth_T [3]. Lazzaretto et al. [4] developed artificial

machine maps using generalized maps taken from literature

with appropriate scaling techniques. Results were

validated with test measurements from real plants. Many

authors have used neural networks approaches. Gustafson

el al. [5] obtained steady-state compressor performance

maps from correlations of transient compressor maps. The

learned correlations were expected to provide information

for predicting stall and surge inception. Yu et al. [6]

predicted an axial compressor's performance map based on

neural networks. Training was performed with

experimental results provided from manufacturers.

Ghorbanian et al. [7] used different types of neural

networks to simulate the performance maps of an axial

compressor; data was obtained from a compressor rig test.

Jiang et al. [8] developed an analytical model for the

centrifugal compressor based on first principles where

energy transfer is taken into consideration. The dynamic

performance including losses was determined from the

compressor geometry. In addition incident and friction

losses were modeled while the clearance, backward and

volute losses were considered constants for all off-design

conditions.Witkowski et al. [9] developed a one-

dimensional analysis for the SR-30 gas turbine components

used in education and research settings. Based on design

point performance, the flow through the radial compressor

and the one stage axial turbine was computed. The

corresponding initial conditions of the flow along with the

dimensions for each component were used. Losses were

estimated using Stodola's equation for slip factor

calculations. These calculations were only presented for a

design speed of 78000 rpm. Studies in developing the

component maps of a SR-30 turbine were performed by

various authors taking into consideration the design point

analysis developed by Witkowski et al. [9]. May et al. [11]

described a procedure to obtain the velocity triangles for

the characteristic maps of the SR-30 gas turbine. An

International Journal of Engineering Research & Technology (IJERT)

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IJERTV10IS030090(This work is licensed under a Creative Commons Attribution 4.0 International License.)

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iterative calculation in Excel was used, but the

mathematical analysis to achieve this goal was not

presented. Resulting maps were plotted using the Smooth-

C and Smooth-T commercial softwares. The model has

been used to monitor health operation of the engine. Based

on the mean-line analysis, Léonard et al. [10] developed

the compressor characteristic curves. The results were

stored in a line format to prevent numerical interpolation

problems. These authors did not use a map for the turbine,

instead, it was characterized using Stodola's ellipsis

equations. Compressor and turbine losses were considered

in the calculation of the velocity triangles. Oh et al. [11]

tested most of the loss models previously published in the

literature. They found an optimum set of empirical loss

models for a reliable performance prediction of centrifugal

compressors. May et al. [12] cites these loss correlations

for the calculation of the compressor map. In radial flow

machines, like centrifugal compressors, the angular

momentum imparted to the flow is reduced by a factor

known as the slip factor. von Backström [13] developed a

method that unifies the trusted centrifugal impeller slip

factor prediction methods of Busemann [14], Stodola [15],

Stanitz [16], Wiesner [20], Eck [17] and Csanady (reported

by Dixon [18]) in one equation. This equation, which is

used in the present work, may be adjusted for specifically

constructed families to improve the accuracy of the

prediction.There is no compressor map or turbine map

available for the SR-30 turbojet engine that has been

validated using experimental data.

The present work develops an analytical model of the SR-

30's component maps. A one-dimensional approach is

applied to perform the calculation at the design point

performance. The final goal is to produce tabulated

functions that may be introduced into an engine model. In

addition, interpolation will be applied to obtain values of

the dependent parameters such as the referred mass flow

rate m , the pressure ratio PR and the efficiency .

II. ANALYTICAL MODEL

In this section, the analytical model of the component maps

is described. The design point and off design performance

are conducted taking into account the limits of the engine

operation. Design point calculations are based on iterative

procedures. This consist on assuming a value of the axial

velocity component, then, performing the calculations

required to find the aero-thermodynamic properties of the

fluid and finally recalculating the assumed velocity.

Iteration is finished when the difference between the initial

guess and the calculated axial flow velocity is within the

imposed tolerance. As a result, the value obtained for the

remaining fluid parameters will be found. Compressor map

and turbine map models are described separately. The

properties of the working fluid in a gas turbine engine have

a powerful impact upon its performance. The working

fluid, which passes through the compressor, is assumed as

dry air and the working fluid that passes through the

turbine is assumed as the product of combustion; the

mixture of dry air and kerosene fuel. Water content due to

humidity is neglected. The correlations used for the fluid

properties were taken from Walsh et al. [19]. The equations

required to find the energy losses in the centrifugal

compressor was obtained from Oh et al. [11], and the loss

coefficients in the turbine, were taken from Dixon [18] and

Cohen et al. [20]. Moreover, the thermodynamic relations

used are derived from the Mollier diagram found in Dixon

[18].

A. Compressor Map

The SR-30 turbojet engine uses a centrifugal

compressor. The analysis for the flow through the

compressor is based in the one-dimensional steady flow

analysis, in which according to Dixon [18], the velocity

and density are regarded as uniform across each section of

a duct or passage. By continuity, the mass flow rate ( m )

through the compressor is given by

222222222111 ACACACACm aDDaDIIaIa ==== (1)

In equation (1), , Ca and A denote the air density, air

axial velocity and the cross-sectional area respectively. The

subscripts 1, 2I, 2D and 2 denote stations corresponding to

the entrance to impeller, exit from impeller, entrance to

vaned diffuser and the exit from vaned diffuser,

respectively. These stations can be visualized in Fig. 1.

Figure 1. Nomenclature of stations used in the compressor

Compressor maps will be developed from data points

obtained from calculations of the performance parameters

such as: mass flow rate m , rotational speed N, pressure

ratio PRc and isentropic efficiency c. For this purpose,

velocity triangles are calculated initially at the design point.

At this point, the mass flow rate and the rotational speed

are 297.0=m kg/s and N = 78000 rpm, respectively. In

addition, the geometric data and input conditions published

by Witkowski et al. [9] are used.

The first velocity triangle, depicted in Fig.2, is

calculated at the impeller inlet. According to this triangle,

the relation between the velocity vectors is given by

Figure 2. Velocity triangle at impeller inlet




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1 1 1= −V C U (2)

where V1 and C1 are the relative and absolute velocity

of the air, respectively. The term U1 denotes the impeller

eye velocity vector or tangential velocity vector. It is

assumed that the air enters the impeller eye in the axial

direction, i.e., 1 1a

C C= . The angle between the relative

velocity and the tangential velocity is 1=57. At the

impeller inlet (known also as eye or inducer), the effective

diameter is defined according to

( )2 2

1 0.5 t rD D D= + (3)

where Dt is the diameter at the impeller eye tip and Dr is

the diameter at the impeller eye root. The annulus area of

impeller eye is

( )2 2

14

t rA D D

= −

(4)

The mean tangential speed at the inducer may be

calculated by

1 1U D N= (5)

In addition, velocities at impeller eye tip (Ut=DtN) and

at impeller eye root (Ur=DrN) may be found. It is

important to check that the velocities at the impeller eye tip

should be less than the speed of sound (Mt< 1) to avoid

breakdown of flow and excessive pressure loss. To

calculate the axial component of the absolute velocity Ca1,

a trial and error process will be performed. Fig.3 shows the

iterative procedure followed to obtain the parameters at

compressor inlet or station 1, according to this flow

diagram, the density is calculated on the basis of the known

stagnation temperature and pressure. From the continuity

equation an initial estimate of the axial velocity Ca1 is

obtained. With this value and with the help of

thermodynamic relations from the Mollier diagram, the

density is recalculated, and the actual velocity is

determined from the continuity equation. If the assumed

and calculated velocities do not coincide, it is necessary to

iterate until convergence is reached.

Figure 3. Flow diagram of the iterative procedure at the impeller inlet

Equations required to calculate Ca1, are presented. The

static enthalpy h1 may be obtained by

1

2

01 1

1

2h h C= −

(6)

where 01

h is obtained from a polynomial function

dependent on the temperature T01, i.e. h01=fh(T01). The

static pressure p1 is given by

01 11

01

R

pp

e −

=

(7)

Here, p01 is the total pressure. The terms 01 and 1 are

functions dependent on the temperature T01 and T1, i.e. 01

=f (T01) and 1 =f (T1). The air density may be calculated

from the equation of state for a perfect gas, thus

1

1

1

p

RT =

(8)

where R is the gas constant for the air and T1 may be

found as a function of the enthalpy h1, i.e. T1 = fh (h1).

Finally, the axial velocity Ca1 is given by

1

1

1

a

mC

A=

(9)

Additional calculations are considered to obtain the

relative velocity V1, 2 2

1 1 1V C U= − and the absolute Mach

number, 1 1absM C RT= .

The velocity triangle at the impeller tip is shown in

Fig.4.




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Figure 4. Velocity triangle at impeller outlet

According to Fig. 4, the air leaves the impeller tip with

an absolute velocity C2I. This velocity has a tangential or

whirl component Cw2I and a small radial component Cr2I ,

they are related by

2

2 2

2 2I w I r IC C C+= (10)

Under ideal conditions the whirl component will be

equal to the impeller tip speed U2I . Due to the relative

eddies in the impeller blade passage the impeller exit flow

angle differs from the impeller exit blade angle; a measure

of deviation can be calculated using the concept of a slip

factor. There are many correlations predicitng the slip

factor. Von Backström [13] describes a method that unifies

the centrifugal impeller slip factor (). The prediction

method of Dixon [18]), that uses one equation, will be

used. This equation is given by

( )

1

1 0.5cos

1

1o Se

F

= −

+

(11)

where Fo is a correction constant corresponding to the

compressor. This value is adjusted to improve the

accuracy of the prediction. The blade angle at impeller exit

is denoted by . The ratio given l/Se is known as the blade

row solidity. The blade length l and blade spacing Se are

determined with ( )e i- / cosl = r r and /e e b= 2 r nS . The

terms ri and re are the radius of the blade leading edge and

blade trailing edge, respectively and the number of blades

is denoted by nb.The area at impeller exit, A2I, and the blade

speed U2I are calculated with 2 2 2I I I

A D b= and 2 2I I

U D N=

respectively. The terms D2I and b2I are the impeller tip

diameter and the axial width at impeller exit respectively.

In terms of the slip factor and the blade tangential speed,

the whirl component of the absolute velocity Cw2I are

determined with

2 2w I IC U= (12)

Some designers consider the value of the radial

component C2I the same as the inlet axial velocity Ca1

(Cohen et al.[20]), but, in the current model it is obtained

with an estimated value based on equation (10). The final

value of the radial component is determined following an

iteration procedure which is shown in Figure 5.

Figure 5. Flow diagram of the iterative procedure at the impeller outlet

According to the diagram shown in Fig. 5, static and

stagnation pressures and temperatures have to be

determined. Thus, the total pressure p02I is given by

02 01 2I Ip p PR= (13)

In equation (13), p01 is given as data and PR2I is the

pressure ratio. Initially PR2I is guessed and then calculated.

The stagnation temperature T02I may be obtained from a

correlation function of the stagnation enthalpy h02I, i.e.

( )0202 II ThT f= . The enthalpy h02I is given by

02

0201

01

2

IsI

I

h hh h

= +

− (14)

where h01 may be obtained from a correlation in

function of the stagnation temperature T01 which is given as

data. The term 2I denotes the efficiency that is initially

guessed and finally calculated after some iterations. The

stagnation enthalpy at an isentropic process h02Is is obtained

by the correlation function ( )0202 IsIs hTh f= . In the same

way, the temperature T02Is may be given by the correlation

function ( )0202 IsIs TT f = . The term 02Is is an entropy

function obtained by the following expression

02

02 01

01

ln IIs

pR

p = +

(15)

where the entropy function 01 is obtained using the

correlation function

( )01 01f T = . The static temperature

T2I is given in terms of the static enthalpy h2I with the




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correlation function ( )22 II ThT f= . The enthalpy h2I is

calculated with the expression

2

2

02 2

1

2I I Ih h C−=

(16)

The static pressure p2I is given by

02 2

202

I II

R

Ipp

e −

=

(17)

here the entropy functions 02I and 2I are given in term

of the temperatures T02I and T2I respectively, i.e.

( )0202 II Tf

= and ( )2 2I If T = .The density and the

radial component of absolute velocity are calculated using

( )22 2 II I RTp = and ( )2 22 I Ir I mC A= respectively. The Mach

numbers should be calculated to ensure that the flow does

not exceed the velocity of sound. The performance

parameters are also affected by losses that occur inside the

compressor. The correlations used for the compressor

losses were taken from Oh et al. [22] and Jiang et al. [8].

The pressure ratio and isentropic efficiency may be given

by

( )1

01

2 1

p

inteulerI

h hPR

c T

−

− = +

(18)

and

2

inteulerI

act

h h

h

− =

(19)

where, heuler is the ideal specific enthalpy change of the

working fluid which depends on the slip factor and the

tangential velocity of the blades at the impeller tip. Thus 2

2euler

Ih U = . The term

intΔh represents the summation

of all internal losses in the impeller such as incidence loss,

blade loading loss, skin friction loss and clearance loss,

thus int i bld f cl

Δh = Δh + Δh + Δh + Δhs

. The actual specific

enthalpy denoted by hact, is the sum of the ideal specific

enthalpy change and the parasitic loss correlations

(hact=heuler+hprs). The parasitic losses, hprs, are

assumed to be the sum of the disc friction loss,

recirculation loss and the leakage loss, i.e. hprs= hdf +

hrc +hlk.

The velocity triangle at the vaned diffuser inlet is

depicted in Fig. 6.

Figure 6. Velocity triangle at the vaned diffuser inlet

In Fig. 6, only the absolute velocity C2D and its

components are present. The relative velocity vanishes

because there is no moving element. To determine the

whirl component Cw2D of C2D, the conservation of the

angular momentum within the vaneless diffuser can be

assumed, i.e. Cw2D r2D = Cw2I r2I . Thus, the whirl

component of the velocity is given by

2 2

2

2

w I I

w D

D

C rC

r= (20)

The area and absolute velocity are obtained from A2D =

2r2D b2D and 2 2

2 22 r D w DDC C C= + respectively. The iterative

procedure performed here is similar to that followed at the

impeller exit. No further energy is supplied to the air after

it leaves the impeller, so that neglecting the

effect of friction, the angular momentum must be constant

(Cohen et al. [20]). The above implies that no work is done

in the vaneless diffuser and the process is adiabatic. Thus,

the following relation is obtained

02 02I D

h h= (21)

The pressure ratio and isentropic efficiency is calculated

in the same way as in the last stage. Although here, hintI is

substituted by hintVD which correspond to the losses in the

impeller and the vaneless diffuser, as follows: hintVD =

hintI +hvld.

Finally, calculations of the velocity triangle at the

diffuser output are performed. Here, the whirl component

of the velocity, is found using conservation of angular

momentum, which yields Cw2 = Cw2D r2D/ r2. The area and

the absolute velocity (magnitude) are A2 = 2 r2 b2 and 2 2

2 22 a wC C C= + respectively. The iterative procedure to

obtain the velocities and the thermodynamic states are

similar to the previous stations. Assumptions done for the

station of vaneless diffuser may be applied for the vaned

diffuser, thus h02D = h02. In the calculations of the pressure

ratio and the isentropic efficiency, the losses through the

entire compressor stage (i.e., impeller, vaneless diffuser

and vaned diffuser) have to be considered. These losses

denoted by hintT may be obtained by

hintT=hintVD+hiD+hsfD, where hintVD represents the

internal losses in the impeller and vaneless diffuser, hiD is

the incidence loss in the diffuser, and hsfD is the skin

friction loss in the diffuser.

The velocity triangles corresponding to the off-design

performance are constructed taking into account the mass




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flow rate and rotational speed in the range of

m = 0 - 0.5 kg/s and N =50000-90000rpm, respectively. In

the off-design calculations it is assumed that the absolute

inlet velocity remains axial and the design point value of

the slip factor was constant for all off-design conditions.

B. Turbine Map

The SR-30 is a single stage axial turbine composed of

one row of nozzles and one rotor. Figure 7 shows the

nomenclature of the stages employed in the calculations; N

refers to the nozzle and R the rotor.

Figure 7. Nomenclature of stations used for the turbine

Ainley and Mathieson correlations [18] were used to

calculate the energy losses. These estimated losses will be

useful in obtaining characteristic parameters such as the

mass flow rate m , the rotational speed N, the pressure ratio

PRt and the efficiency t. The operating conditions used in

establishing the design point velocity triangles were:

m = 0.3 kg/s and N=78000rpm. A mean annulus diameter

is assumed in the calculations. This approach is valid when

the ratio of the tip radius rt to the root radius rr is low (rt/rr

= 1:38). The turbine considered in this work meets this low

aspect ratio. Moreover, we can consider the mean diameter

equal in all stations because the difference between the

station mean diameters are negligible. Thus, the effective

mean diameter may be given by ( )2 2

m r tD = 0.5 D D+ . With the

effective diameter and the rotational speed, the mean blade

speed is U = DmN. The mean annulus area is calculated

by ( )2 2

r tmA = π D - D / 4 .

Geometric dimensions and inlet conditions were

extracted from Witkowski et al. [9]. By using the geometric

measurements of the turbine, a tip clearance of 0.54mm

was obtained. It is assumed that the stator blades have the

same shape as the rotor blades. The fluid through the

turbine is the air-fuel combustion products.

Fig. 8 depicts the velocity configurations at each station

in this turbine. In a single-stage turbine the velocity at

turbine inlet C3 will be axial (Cohen et al. [20]), thus, C3 =

Ca3 (Ca3 is the axial component of the velocity at turbine

inlet or station 3). The value of this velocity will be taken

as constant for all off-design conditions.

Figure 8. Velocity diagram of the axial turbine stage

Fig.9 shows the iterative procedure to calculate Ca3.

This procedure is similar to the calculations made at

compressor inlet.

Figure 9. Flow diagram of the iterative procedure at the nozzle inlet

Initially, the velocity Ca3 is guessed. Then, the static

temperature is found using a correlation in terms of the

static enthalpy h3, i.e. ( )3 T 3T = f h . The enthalpy h3 is

obtained by

3 03

2

3

1h = h - C

2 (22)

where: h03 = fh(T03, FAR) and FAR is the fuel-air ratio. The

static pressure is obtained using




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03 33

03

R

pp

e −

=

(23)

where 03=f(T03) and 3=f(T3). Similarly to previous

cases, the density is calculated with 3=p3/(RT3). From the

continuity equation, the axial velocity Ca3 is calculated by

a3 3 mC = m / (ρ A ) . Finally, the corresponding mach number

isdetermined using 3 33abs

C RTM = .

Fig. 8 shows the velocity triangle at the nozzle outlet or

station 4N. It is assumed that the nozzle angle (4N) and the

angular momentum at outlet remains constant. Therefore,

for all off-design conditions the deviation of flow across

the nozzle remains constant, thus 4N=constant. As a

result, the relationship between the whirl component Cw4N

and the axial component Ca4N (notation shown in Fig.8)

will be constant according to the relation.

4

4

4

1tan w N

N

a N

C

C

−=

(24)

Since no work is done (adiabatic process) by the gas in the

nozzle, the stagnation enthalpy must be constant over the

annulus at the outlet. Thus, h04N = h03. Fig.10 shows the

flow chart used for the iterative procedure to obtain the

axial component of the absolute velocity Ca4N.

Figure 10. Flow diagram of the iterative procedure at the nozzle outlet

According to this flowchart, the absolute velocity Ca4N

is guessed. Using this value and the angle 4N, the velocity

component Cw4N is obtained (Cw4N=Ca4Ntan4N). The

absolute velocity C4N is given by 2 2

4N a4N w4NC = C + C . The

static temperature T4N is found in term of the enthalpy, i.e.

( )44 NN ThT f= . The static enthalpy h4N is given by

4

2

04 4

1

2N N Nh h C−=

(25)

The static pressure may be calculated by

03 44

03

NsN

R

pp

e −

=

(26)

Where, ( )03 03,f T FAR = , ( )4 4 ,Ns Nsf T FAR = , and

( )4 4 ,Ns T NsT f h FAR= . The enthalpy under isentropic process

h4Ns is found using

4

2

44

2Ns

NN N

Ch h −=

(27)

where the total loss coefficient in the nozzle is denoted by

N. Neglecting the effect of friction, the critical pressure

ratio is

03

11

2c

p

p

− +=

(28)

If P03/P4N<P03/Pc, then the nozzle is not choked, otherwise,

it is chocked. The density 4N and the axial flow velocity

Ca4N are as follow 4N=p4N/(RT4N)

anda4N 4N mC = m / (ρ A )& . The absolute velocity and

relative flow angle are ( ) 2

44 4

2

4 a NN w NV C U C= − + and

4N=tan-1[(Cw4N-U4)/Ca4N].

The velocity triangle at rotor outlet or station 4 is depicted

in Fig. 8. The flow diagram of the iterative procedure at

this station is presented in Fig. 11.

Figure 11. Flow diagram of the iterative procedure at the rotor outlet




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According to this diagram, the exit angle 4 is constant

and it is calculated from the equations recommended by

Ainley and Mathieson (Dixon [18]), thus

441

11.5 1.154 coss

s e

− = − + +

(29)

In equation (29), is the throat width, s is the pitch and e is

the mean radius of curvature of the blade suction surface.

Equation (29) is for low outlet Mach numbers, i.e. 0 <

M4<0.5. If 4 1M the angle 4 is calculated using

( )41

cos s−

= . More details about this angle can be found

in Dixon [18]. Initially, the axial component of the absolute

velocity Ca4 is guessed using the equality Ca4=Ca4N. The

whirl component of the absolute velocity Cw4 may be

evaluated by

4 4 4 4tanw aC C U= − (30)

The absolute and relative velocity can be found similarly as

in the other stations. The static temperature is found by the

correlation ( )44 ,T

h FART f= . The static enthalpy h4 is given

by

4

2

404

2

Ch h −=

(31)

where h04 is calculated by

( )04 03 4 4 44

4 4

2tan tana N N

a

a N a N

U Ch h UC

C C+= − − (32)

In equation (32), the stagnation enthalpy h03 is obtained by

( )0303 ,h

T FARh f= .Using the stagnation enthalpy h04, the

stagnation temperature is computed with ( )0404 ,T

T FART f= .

The stagnation pressure is found from

0403

t

pp

PR=

(33)

where PRt is the total pressure ratio which is guessed

initially and then calculated by PRt=(03-04ss)/R . The

entropy function 03 is obtained using ( )0303 ,T FARf

= .

The stagnation entropy function 04ss is given by

( )0404 4 4ln

ss ssR p p+ = . The static entropy function 4ss

is obtained from the correlation ( )44 ,ssss T FARf

=

where the static temperature T4ss is calculated using the

correlation ( )44 ,ssss Th FART f= . The static enthalpy h4ss is

given by h4ss=h4s-(T4/T4N)(h4N-h4Ns). The static isentropic

enthalpy h4s is found using

4

2

44

2s R

Vh h −=

(34)

where R is the total loss coefficient in the turbine

rotor.The static pressure p4 was calculated with

04 44

04

R

pp

e −

=

(35)

here ( )0404 ,T FARf

= and ( )44 ,T FARf

= . The isentropic

efficiency is determined using

03 04

03 04

t

ss

h h

h h

−

−

=

(36)

The density and the axial velocity component are

calculated using equations similar to the previous cases. In

addition, the mach number M4 may be found by

4 44M = V RT . After doing the design point performance

analysis, the off-design performance is obtained varying

the speed and the mass flow rate. The speed was varied

from 50000rpm to 90000rpm and the mass flow rate was

varied until the design value was reached (0.3kg/s). The

deviation of flow across the nozzle is assumed as constant

at all off-design conditions. Because 4N is constant the

losses across the nozzle remain constant for all off-design

conditions.

III. RESULTS A. Compressor Map

Results of pressure, temperature, absolute velocity and Mach number in the behavior of the design point are shown in Table I. The values presented here correspond to each station in the compressor. The analysis was generated under the rotational speed of 78000rpm with a mass flow rate of 0.297kg/s. The abbreviation, PW, means present work. The table shows that the highest stagnation pressure is reached at the impeller outlet (P0 = 323kPa).Thepressure at the diffuser outlet is higher than the impeller inlet due to irreversibility. The compression work done by the impeller is accompanied by an increase in temperature, thus, the stagnation temperature at the impeller outlet is 446K. This temperature remains constant through the diffuser. Static pressure and static temperature gradually increase through the compressor. Thus, its maximum values are reached at the diffuser outlet, these are: 249kPa and 429K. The Mach number corresponding to the absolute velocity is maximum, and less than one, at the impeller outlet.




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Table I. Design pointperformance for the compressor

Parameter

Impeller inlet Impeller outlet

Diffuser inlet

Vaned

diffuser

outlet

1 2I 2D 2

Ref. [9]

PW Ref. [9]

PW Ref. [9]

PW Ref. [9]

PW

P0 (kPa) 100 100 323 323 319 307 269 287

T0 (K) 300 300 445 446 445 446 445 446

P (kPa) 93.6 93.6 167 183 200 191 269 249

T (K) 294 294.4 379 380 390 390 430 429

C (m/s) 106 106.3 366 366 335 337 176 188

Mabs 0.31 0.31 0.87 0.94 0.86 0.85 0.42 0.45

The results obtained correlate well with Witkowski et al.[9]. Calculations at impeller inlet, 1, are practically the same as those presented in [9]. Result differences in locations 2I, 2D, 2 are due to the correlations considered in the present work. The highest percent difference, corresponding to the static pressure at the impeller outlet, obtained when compared to [9] was 9.6%. This is most likely due to the difference in the energy loss and slip factor calculations. In the present work, correction factors were used in the energy loss calculations. The slip factor formula used in the present work was different from that used in Witkowskiet al. [9]. Furthermore, the coefficient F0 of the slip factor used was adjusted to obtain an appropriate value.

By varying the mass flow rate (0-0.55 kg/s) and the rotational speed (50,000 rpm - 90,000 rpm), the off-design performance was obtained. Fig. 12 shows the effect of variation of mass flow and rotational speed on the pressure ratio.

Figure 12. Compressor map: Pressure ratio vs mass flow rate

Solid curve lines shown in Fig. 12 are the rotational speed, these are indicated by dimensionless values that vary from 0.64 (50,0000 rpm) to 1.15 (90,000 rpm). These speed lines are obtained by joining pairs of points formed by the pressure ratio and the mass flow rate. The ends of each speed line coincide with the minimum and maximum mass flow rate. Joining imaginarily the minimum mass flow of each speed line it is obtained the surge line, this line is nearly at the left of the maximum efficiency line. Surging is associated with a sudden drop in delivery pressure, and with violent aerodynamic pulsation which is

transmitted throughout the whole machine. The line joining the maximum mass flow points for each rotational speed is called the choking line, that is, no further increase in mass flow is possible beyond this line.

Figure 13 shows the effect of variation of the rotational speed and the mass flow rate on the isentropic efficiency. Curved lines represent the rotational speed, indicated by dimensionless values. Speed lines values are the same as in Fig. 12. The operating point of maximum efficiency on each speed line is approximately the same. Ideally, the compressor should run on the curve connecting these points.

Figure 13. Compressor: Efficiency vs mass flow rate

In Fig. 12, eleven dashed lines intersect the speed lines

at a single point, these are called beta lines. Information on

mass flow, pressure ratio and efficiency is extracted from

each of these points. This information is taken in tabular

form, like Tables II, III and IV. These tables may be

introduced in a SR-30 turbojet engine model, this

procedure is presented in another work. Values of mass

flow rate, pressure ratio and isentropic efficiency of the

compressor are shown in tables II, III and IV, respectively.

The first column of these tables shows the enumeration of

beta lines. In the second row of each table, the percentage

values of the compressor rotational speeds are shown.

Table II. Tabulated values of the compressor mass flow rate.

β Rotational speed, N (%)

64 69 74 79 85 90 95 100 105 110 115

0 0.11 0.12 0.14 0.16 0.18 0.19 0.21 0.23 0.25 0.26 0.28

1 0.12 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31

2 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31 0.33

3 0.15 0.17 0.19 0.21 0.23 0.25 0.28 0.30 0.32 0.34 0.36

4 0.17 0.19 0.21 0.23 0.26 0.28 0.30 0.32 0.35 0.37 0.39

5 0.19 0.21 0.24 0.26 0.28 0.31 0.33 0.35 0.38 0.40 0.42

6 0.22 0.24 0.27 0.29 0.31 0.34 0.36 0.38 0.40 0.43 0.45

7 0.26 0.28 0.30 0.32 0.34 0.36 0.39 0.41 0.43 0.45 0.47

8 0.30 0.32 0.34 0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49

9 0.33 0.35 0.37 0.38 0.40 0.42 0.44 0.45 0.47 0.49 0.50

10 0.37 0.38 0.40 0.41 0.43 0.44 0.46 0.47 0.49 0.50 0.52

Table III. Tabulated values of the compressor pressure ratio


64 69 74 79 85 90 95 100 105 110 115

0 1.68 1.68 1.68 1.67 1.66 1.63 1.60 1.56 1.50 1.44 1.37

1 1.81 1.82 1.82 1.81 1.80 1.77 1.73 1.68 1.62 1.55 1.46

2 1.98 1.99 1.99 1.97 1.95 1.92 1.88 1.82 1.74 1.65 1.55

3 2.16 2.17 2.17 2.16 2.14 2.10 2.05 1.98 1.90 1.79 1.66

4 2.37 2.38 2.38 2.37 2.34 2.30 2.24 2.16 2.06 1.93 1.78

5 2.62 2.63 2.62 2.60 2.57 2.52 2.45 2.37 2.25 2.11 1.93

6 2.90 2.90 2.89 2.87 2.83 2.77 2.70 2.60 2.46 2.30 2.09

7 3.21 3.21 3.19 3.17 3.12 3.06 2.98 2.88 2.75 2.57 2.27

8 3.56 3.55 3.53 3.50 3.45 3.39 3.30 3.20 3.07 2.86 2.40

9 3.95 3.94 3.91 3.88 3.83 3.77 3.69 3.59 3.42 3.05 2.51

10 4.38 4.37 4.34 4.30 4.26 4.19 4.11 3.96 3.67 3.19 2.65




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Table IV. Tabulated compressor efficiency values.


64 69 74 79 85 90 95 100 105 110 115

0 75.5 77.7 78.5 78.2 76.9 74.6 71.3 67.0 61.5 54.7 0.46

1 75.4 77.7 78.7 78.6 77.5 75.5 72.5 68.5 63.3 56.7 0.49

2 76.2 78.1 79.0 78.8 77.7 75.7 72.8 68.9 63.8 57.3 0.49

3 76.4 78.3 79.1 79.0 78.1 76.2 73.5 69.8 64.9 58.6 0.51

4 77.9 79.6 79.3 79.2 78.3 76.5 74.0 70.3 65.6 59.4 0.53

5 77.6 79.0 79.5 79.3 78.3 76.7 74.2 70.8 66.3 60.4 0.53

6 78.3 79.3 79.6 79.3 78.3 76.7 74.4 71.2 66.8 61.1 0.53

7 79.0 79.5 79.6 79.2 78.3 76.8 74.8 72.1 68.5 63.5 0.54

8 79.4 79.6 79.5 79.0 78.1 76.9 75.1 72.9 70.0 64.9 0.53

9 79.5 79.5 79.3 79.0 78.2 77.2 75.9 74.0 70.7 63.2 0.51

10 79.3 79.3 79.1 78.8 78.2 77.3 76.1 73.8 70.0 60.2 0.49

Figure 14 compares the results obtained in the present work with those presented by May et al. [12]. The last one, indicated in the legend as paper. Three speed lines of the compression ratio versus mass flow rate graph were compared. The dimensionless values of these rotational speeds are 0.7 (54,000 rpm), 1 (78,000 rpm), and 1.1 (90,000 rpm). Results show a better match as speed increases. In the present work, when the compressor operates near the lowest rotational speed, the choking point is reached with a lower mass flow rate. As speed increases, results of speed lines between the two works match better.

Figure 14. Comparison of results for the compressor map.

B. Turbine Map

Table V shows comparative results between the present work, indicated by PW, and those presented by Witkowski et al. [9], indicated as Ref. [9]. Results are obtained in the design-point performance, that is, at a rotational speed of 78,000 rpm and a mass flow rate of 0.3 kg/s. As shown in the table, static and stagnation pressure decrease across each turbine location. The temperature is highest at the stator inlet. At this location, in the present work, the stagnation temperature is 973K while the static temperature is 967K. The Mach number reaches its maximum value at the stator output, this value is less than 1 (near the transonic region, M=1) in both results. If M=1,choking will occur. Both results match closely at locations 3 and 4N. It was evidenced that Witkowski et al. [9] has a mistake in the results of static pressure and static temperature, at location

3. A great difference between both results is observed in the absolute and relative velocities, at location 4. This may be due to the tip clearance value used to calculate the tip clearance loss. In the present work a value of 0.54 mm was used, this value was obtained by measurement carried out in the compressor of the experimental module. In Witkowski et al. [9] a value of 2.44 mm was used.

Table V. Design point performance for the turbine

Parameter

Stator inlet Stator outlet Rotor outlet

3 4N 4

Ref. [9]

PW Ref. [9]

PW Ref. [9]

PW

P0 (kPa) 250 250 234 229 120 120

T0 (K) 973 973 973 973 841 855

P (kPa) 94 244 128 128 115 106

T (K) 294 967 834 841 832 827

C (m/s) 114 114 557 550 141 250

Mabs 0.19 0.19 0.98 0.97 0.25 0.37

Off-design performance results are shown in Figures 15 and 16. Figure 15 shows the isentropic efficiency and mass flow rate against the work parameter, DH/T = (h4 - h3)/T3. In Fig. 15, each line represents the rotational speed. Colors identify the value of each of these speeds. As the speed increases, the maximum efficiency is reached at a higher value of the working parameter. For each speed, the efficiency remains approximately constant above a certain value of the working parameter. This behavior is due to the fact that the loss coefficients have a small variation in a wide range of incidence. Furthermore, this approximately constant efficiency is greater as the rotational speed increases.

Figure 155. Isentropic efficiency against work parameter.

In Fig. 16 it is observed that for each line of rotational speed, the maximum mass flow increases as the working parameter increases. An increase in mass flow occurs until choking conditions are reached. For all speed lines, choking occurs at the same mass flow rate and at a value of the working parameter greater than 100J-kg/K.




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Figure 166. Mass flow rate against workparameter.

In Fig. 17, results of the present work were compared

with those presented by May et al. [12], the latter named in the legend as paper.For this comparison, the relationship between mass flow rate and compression ratio was used. In this relationship, graphs of speed lines corresponding to 78,000 rpm are shown. The solid blue line is the one obtained in the present work while the red dashed line is the one obtained in May et al. Graphics are similar only in shape.The difference in the numerical values between both graphs is great. This is due to the fact that the maximum flow taken into account in May et al. [12] is close to 0.22 kg/s, while in this work 0.3 kg/s was considered. The value of 0.3 kg/s is more reasonable, since it represents the combustion mass flow that would be the sum of the air mass flow coming from the compressor plus the fuel mass flow entering the combustion chamber.

Figure 177. Comparison of results for the turbine map.

Tabulated values of mass flow rate and isentropic efficiency of the turbine are shown in tables VI and VII, respectively. The first column of these tables shows the values of the working parameter (DH/T or simply DHT). The second row of each table shows the percentage values of the rotational speeds of the turbine, these values are the same as for the compressor.

Table VI. Tabulated values of the turbine mass flow rate.

DHT Rotational speed, N (%)

64 69 74 79 85 90 95 100 105 110 115

10 0.17 0.18 0.19 0.19 0.20 0.21 0.22 0.22 0.23 0.24 0.24

25 0.21 0.22 0.22 0.22 0.23 0.23 0.24 0.24 0.25 0.25 0.26

40 0.24 0.25 0.25 0.25 0.25 0.25 0.26 0.26 0.27 0.27 0.27

55 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.28 0.28 0.28 0.28

70 0.29 0.29 0.29 0.28 0.28 0.28 0.29 0.29 0.29 0.29 0.29

85 0.30 0.30 0.30 0.29 0.29 0.29 0.29 0.29 0.29 0.30 0.30

100 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

115 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

130 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

140 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

150 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

Table VII. Tabulated values of the turbine efficiency.

DHT Rotational speed, N (%)

64 69 74 79 85 90 95 100 105 110 115

10 74.4 72.8 71.1 69.2 67.3 65.4 63.4 61.4 59.5 57.5 0.56

25 78.3 78.4 78.2 77.8 77.2 75.8 75.7 74.8 73.9 72.8 0.72

40 76.5 77.4 78.0 78.3 78.4 78.2 78.2 78.0 77.5 77.0 0.76

55 73.6 75.2 76.3 77.2 77.8 78.4 78.4 78.5 78.5 78.4 0.78

70 70.6 72.6 74.2 75.4 76.4 77.7 77.7 78.1 78.4 78.6 0.79

85 67.5 69.9 71.9 73.5 74.8 76.7 76.7 77.3 77.8 78.2 0.78

100 64.4 67.2 69.4 71.4 73.0 75.4 75.3 76.2 76.9 77.5 0.78

115 62.3 64.4 67.0 69.2 71.1 74.0 74.0 75.0 75.8 76.6 0.77

130 62.3 64.3 66.1 67.7 69.2 72.4 72.4 73.7 74.7 75.6 0.76

140 62.3 64.3 66.1 67.7 69.2 71.7 71.7 72.8 74.0 75.0 0.76

150 62.3 64.3 66.1 67.7 69.2 71.7 71.7 72.8 73.8 74.7 0.75

IV. CONCLUSION

In this work, a one-dimensional analytical model of the compressor and turbine maps was made. The unified slip factor formula recommended by von Backström [13] was used. This formula was used in obtaining the whirl component of the absolute flow velocity at the outlet of the compressor impeller. In the compressor map calculations, the internal and parasitic loss correlations selected by Oh et al. [22] were used. In both maps calculations, iteration loops were used to find the axial component of the absolute flow velocity. Many equations were obtained from the Mollier diagram of the corresponding component. Finally, tabulated forms of the compressor and turbine maps were obtained. The results of the design point calculations for each component were compared with that presented by Witkowski et al. [9], a significant match was observed. Off-design performance results were compared with those obtained by May et al. [12]. These match well in the case of the compressor, turbine results showed similar behavior with a significant numerical values differing. This is most likely due to fact that May et al. [12] used a maximum flow rate of 0.22 kg/s while the calculation performed used a value of 0.3 kg/s. This value represents the combustion mass flow which would be the sum of both the air mass flow and the fuel mass flow entering the combustion chamber.

In order to further improve the modeling of the component maps, the validation should be developed by using experimental data or with maps obtained from manufacturers. In addition as built construction details should be used when ever available.

ACKNOWLEDGEMENTS:

The authors would like to acknowledge the interest and

encouragement of Dr. Allan Valponi. Views express in

this paper are those of the authors, and neither of persons

acknowledge herein nor of any funding source nor of their

institutions.




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analytical modeling of characteristic maps of the sr-30

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